Text: Allen Hatcher: Algebraic Topology, available at http://www.math.cornell.edu/~hatcher/.
Grader: David Sheppard. Email sheppard@math.mit.edu to set up an appointment.
The plan is to first cover the basic theory of the fundamental group and covering spaces following Allen Hatcher, Algebraic Topology, chapter 1. The remaining time will be spent on more specialized topics. We meet two times a week. Each time we will have two 25 minutes lectures.
Here is the break down of the individual lectures:
Lecture 1: Discuss homotopy of paths, composition of paths, and define the fundamental group. State and prove proposition 1.2.
Lecture 2: State and prove proposition 1.3, and, if time permits, propositions 1.5 and 1.6.
Lecture 3: Prove statements (a) and (b) of the proof of theorem 1.7.
Lecture 4: State and prove theorem 1.7.
Lecture 5: State and prove theorem 1.8. Why is this an important theorem?
Lecture 6: State and prove theorem 1.9. Give examples.
Lecture 7: State and prove theorem 1.10 and corollary 1.11.
Lecture 8: Discuss the cartesian product and induced homomorphisms. Prove proposition 1.12.
Lecture 9: Prove proposition 1.14 and corollary 1.16.
Lecture 10: Discuss retracts, deformation retracts; prove proposition 1.17.
Lecture 11: State and prove proposition 1.18.
Lecture 12: The construction of the free product of groups from page 41. (Disregard the discussion of ``sum'' and ``product'' at the top of the page.)
Lecture 13: Free groups; the universal property of the free product; Z/2 * Z/2.
Lecture 14: State the Seifert-van Kampen theorem (see James R. Munkres: Topology, Theorem 70.1). Discuss some examples.
Lecture 15: Begin the proof of the Seifert-van Kampen theorem (see James R. Munkres: Topology, Theorem 70.1).
Lecture 16: Finish the proof of the Seifert-van Kampen theorem.
Lecture 17: Linking of circles (ex. 1.23), and, if time permits, the Hawaiian earrings (ex. 1.25).
Lecture 18: Define knots. Discuss torus knots following ex. 1.24.
Lecture 19: Give the definition of a CW-complex on page 5, and discuss as many of the examples on page 6 as possible.
Lecture 20: State and prove proposition 1.26.
Lecture 21: Discuss the CW-structure on an oriented surface of genus g. State and prove corollary 1.27. If time permits, treat non-orientable surfaces, too.
Lecture 22: State and prove corollary 1.28 and give plenty of examples.
Lecture 23: Give the definition of a covering space; give examples. (Coordinate your lecture with the person who give lecture 24.)
Lecture 24: Show that the wedge of two circles has a covering space that is simply-connected. Discuss some of the examples on page 58. (Coordinate your lecture with the person who give lecture 23.)
Lecture 25: State and prove propositions 1.30, 1.31, and 1.32.
Lecture 26: State and prove propositions 1.33 and 1.34.
Lecture 27: Discuss the construction of the universal covering on pages 63-65.
Lecture 28: Explain example 1.35. Try to give rigorous arguments.
Lecture 29: Cover propositions 1.36 and 1.37.
Lecture 30: State and prove (the remaining part of) theorem 1.38. Illustrate the theorem by examples.
Lecture 31: Discuss deck transformations and prove proposition 1.39.
Lecture 32: Discuss group actions and prove proposition 1.40. Give examples.
Lecture 33: Discuss group actions on spheres following example 1.43. If time permits, treat also example 1.44.
Lecture 34: Discuss Cayley complexes.
Problem sets:
Problem set 1: Due Thursday, February 27. The problems are all from the exercises on pages 38-39 of the text. Please hand in solutions of problems 5, 6, 10, 15, and 16.
Problem set 2: Due Thursday, March 20. The problem set is here.